Hartle-Hawking state, factorization and tensor network in 3d gravity

In this talk, I will first describe work on 3d quantum gravity with two asymptotically AdS regions, in particular, using its relation with coupled Alekseev-Shatashvili theories and Liouville theory. Expressions for the Hartle-Hawking state, thermal 2n-point functions, torus wormhole correlators and Wheeler-DeWitt wavefunctions in different bases are obtained using the ZZ boundary states in Liouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity are uplifted to 3d gravity, with two copies of Liouville theory in 3d gravity playing a similar role as Schwarzian theory in JT gravity. The connection between 3d gravity and the Liouville ZZ boundary states are manifested by viewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormhole boundaries glued along the ZZ boundaries. I will also introduce the factorization problem of the Hartle-Hawking state in this setup. With the relevant defect operator that imposes the necessary topological constraint for contractibility, the trace formula in gravity is modified in computing the entanglement entropy. This trace matches with the one from von Neumann algebra considerations, further reproducing the Bekenstein-Hawking area formula from entanglement entropy. I also propose a calculation for off-shell geometrical quantities that are responsible for the ramp behavior in the late time two-point functions, which follows from the understanding of the Liouville FZZT boundary states in the context of 3d gravity, and the identification between Verlinde loop operators in Liouville theory and "baby universe" operators in 3d gravity. Lastly, I will describe ongoing work where we use the universal boundary OPE data in 2d CFT to construct a real space renormalization group flow procedure and determine its eigenstate. 3d geometry emerges from the algebraic data via the form of an exact tensor network. The talk will be based on: 2309.05126 and some ongoing work.